Heavy config geo involving mixtilinear
by Assassino9931, Mar 30, 2025, 1:23 PM
Let 
be an acute-angled triangle 
with 
and incenter 
. Let 
be the mixtilinear circle at vertex 
, i.e. the circle internally tangent to the circumcircle of 
and also tangent to lines 
and 
. A circle 
passes through points 
and 
and is tangent to at point 
, with 
and being inside 
. Prove that:
be an acute-angled triangle 
with 
and incenter 
. Let 
be the mixtilinear circle at vertex 
, i.e. the circle internally tangent to the circumcircle of 
and also tangent to lines 
and 
. A circle 
passes through points 
and 
and is tangent to at point 
, with 
and being inside 
. Prove that:
This post has been edited 2 times. Last edited by Assassino9931, 5 hours ago
be an acute scalene triangle. A point 
varies on its side 
and 
are the midpoints of the arcs 
and 
(not containing 
and 
,
passes through a fixed point, independent of the choice of 
lying outside the circle 
,
and 
.
is chosen on the segment 
.
and 
be the midpoints of segments 
and 
respectively. The circumcircle of triangle 
intersects 
)
passes through the centroid of triangle 
.
has circumcircle 
,
to 
and meet 
,
to 
meets 
.
are positive integer.
and 
with 
,
and 
,
,
and 
.
the midpoint of the arc 
and 
intersect the perpendicular bisectors of 
and 
,
intersect at 
.
a number from a set 
is written --- in the first row numbers 
,
and so on. Exactly 
of them have been chosen, no two from the same row or the same column. Let us denote by 
a number chosen from row number 
.![\[ \frac{1^2}{a_1}+\frac{2^2}{a_2}+\ldots +\frac{n^2}{a_n}\geq \frac{n+2}{2}-\frac{1}{n^2+1}\]](http://latex.artofproblemsolving.com/a/d/5/ad5ae6861da79a08d32093f63827016868856a3a.png)
be the set of positive rational numbers. Construct a function 
such that![\[ f(xf(y)) = \frac {f(x)}{y}
\]](http://latex.artofproblemsolving.com/5/9/1/59172e31cf6b7d885295f6e1daf6891a71739247.png)
,
in 
March 2025
February 2025